In the limit of rapid passage in a time-dependent two-level system of the Landau–Zener type, the initial state undergoes a very simple yet non-trivial rotation. This effect takes place when the Hamiltonian reaches the stationary point of the system. We explain the origin of this rotation by simple means and extend it to open non-Hermitian (NH) systems where new features appear. In addition, we find that in contrast to the Hermitian case, the point at which this simple rotation takes place is not necessarily the point of minimal energy separation. Moreover, in NH Hamiltonian the stationary point may lie in the complex time plane, and then a surprisingly strong correction to the rotation angle appears. The NH aspects of this rotation can be observed in optical and quantum systems where decay rates can be controlled. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
In the limit of rapid passage in a time-dependent two-level system of the Landau–Zener type, the initial state undergoes a very simple yet non-trivial rotation. This effect takes place when the Hamiltonian reaches the stationary point of the system. We explain the origin of this rotation by simple means and extend it to open non-Hermitian (NH) systems where new features appear. In addition, we find that in contrast to the Hermitian case, the point at which this simple rotation takes place is not necessarily the point of minimal energy separation. Moreover, in NH Hamiltonian the stationary point may lie in the complex time plane, and then a surprisingly strong correction to the rotation angle appears. The NH aspects of this rotation can be observed in optical and quantum systems where decay rates can be controlled. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time +T such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.
Non-Hermitian quantum systems with explicit time dependence are of ever increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points of Nth order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behavior. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behavior can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
While the Hamiltonians used in standard quantum mechanics are Hermitian, it is also possible to extend the theory to non-Hermitian Hamiltonians. Particularly interesting are non-Hermitian Hamiltonians satisfying parity–time ( P T ) symmetry, or more generally pseudo-Hermiticity, since such non-Hermitian Hamiltonians can still exhibit real eigenvalues. In this article, we present a quantum simulation of the time-dependent non-Hermitian non- P T -symmetric Hamiltonian used in a pseudo-Hermitian extension of the Landau–Zener–Stückelberg–Majorana (LZSM) model. The simulation is implemented on a superconducting processor by using Naimark dilation to transform a non-Hermitian Hamiltonian for one qubit into a Hermitian Hamiltonian for a qubit and an ancilla; postselection on the ancilla state ensures that the qubit undergoes nonunitary time evolution corresponding to the original non-Hermitian Hamiltonian. We observe properties such as the dependence of transition probabilities on time and the replacement of conservation of total probability by other dynamical invariants in agreement with predictions based on a theoretical treatment of the pseudo-Hermitian LZSM system. Published by the American Physical Society 2024
Non-Hermitian quantum systems with explicit time dependence are of ever-increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points of N-th order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behaviour. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behaviour can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
The main motivation behind the call for this special issue was to gather recent results, developments and open problems in quantum physics with non-Hermitian operators. There have been previous special issues in this journal [1, 2] and elsewhere on this subject. The intention of this issue is to reflect the current state of this rapidly-developing field. It has therefore been open to all contributions containing new results on non-Hermitian theories that are explicitly -symmetric and/or pseudo-Hermitian or quasi-Hermitian. In the last decade these types of systems have proved to be viable self-consistent physical theories with well defined unitary time-evolution and real spectra. As the large number of responses demonstrates, this is a rapidly evolving field of research. A consensus has been reached regarding most of the fundamental problems, and the general ideas and techniques are now readily being employed in many areas of physics. Nonetheless, this issue still contains some treatments of a more general nature regarding the spectral analysis of these models, in particular, the physics of the exceptional points, the breaking of the -symmetry, an interpretation of negative energies and the consistent implementation of the WKB analysis. This issue also contains a treatment of a scattering theory associated with these types of systems, weak measurements, coherent states, decoherence, unbounded metric operators and the inclusion of domain issues to obtain well defined self-adjoint theories. Contributions in the form of applications of the general ideas include: studies of classical shock-waves and tunnelling, supersymmetric models, spin chain models, models with ring structure, random matrix models, the Pauli equation, the nonlinear Schrödinger equation, quasi-exactly solvable models, integrable models such as the Calogero model, Bose–Einstein condensates, thermodynamics, nonlinear oligomers, quantum catastrophes, the Landau–Zener problem and pseudo-Fermions. Applications close to experimental realization are proposed in optics, including short light pulse models, waveguides and laser systems, and also in electronics. We hope that this issue will become a valuable reference and inspiration for the broader scientific community working in mathematical and theoretical physics. References [1] Fring A, Jones H F and Znojil M (ed) 2008 J. Phys. A: Math. Theor. 41 240301 [2] Geyer H, Heiss D and Znojil M (ed) 2006 J. Phys. A: Math. Gen. 39 9963
This special issue contains contributions presented at the workshop on The Physics of Non-Hermitian Operators held at Stellenbosch, South Africa in November 2005. A few additional articles are included from researchers who could not attend the meeting. The workshop was the fourth meeting in a series on Pseudo-Hermitian Hamiltonians in Quantum Physics which commenced in Prague in 2003 as a result of growing interest in non-Hermitian Hamiltonians in general, and PT-symmetric quantum mechanics and quantum field theory in particular. Naturally, Hamiltonians of open systems are an obvious part of this topic, but the focus here is rather on systems where unitary time evolution as such is not undermined. This rather new, and perhaps ambitious, field thus explores a possible generalization of quantum mechanics which incorporates non-Hermitian Hamiltonians into a consistent quantum mechanical framework. New results of a physical and mathematical nature have brought about appreciable progress; earlier work has been uncovered or even rediscovered and links between sub-fields have been established. This issue should provide a stimulus for initiating further advances in the field, re-focus on earlier perspectives and explore open and controversial questions. While non-Hermitian operators have been encountered in theoretical physics since the advent of quantum mechanics, their systematic mathematical investigation and their possible physical implications is a relatively new subject, further explored in this issue. We would like to thank the University of Stellenbosch and the South African National Research Foundation for their financial support, without which it would not have been possible to plan and organize the workshop. Organizational aspects and technical support were very ably taken care of by Christine Ruperti and Tinus Botha whom we would like to thank again. Finally, we also appreciate and acknowledge the invitation from the Editorial Board of Journal of Physics A: Mathematical and General for the opportunity to publish this special issue and trust that this will indeed widen the interest and stimulate progress in the field.